34
MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE J OSÉ L UIZ B OLDRINI MARKO ROJAS -MEDAR On the convergence rate of spectral approximation for the equations for nonhomogeneous asymmetric fluids Modélisation mathématique et analyse numérique, tome 30, n o 2 (1996), p. 123-155 <http://www.numdam.org/item?id=M2AN_1996__30_2_123_0> © AFCET, 1996, tous droits réservés. L’accès aux archives de la revue « Modélisation mathématique et analyse numérique » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impres- sion systématique est constitutive d’une infraction pénale. Toute copie ou im- pression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

JOSÉ LUIZ BOLDRINI

MARKO ROJAS-MEDAROn the convergence rate of spectral approximation forthe equations for nonhomogeneous asymmetric fluidsModélisation mathématique et analyse numérique, tome 30, no 2(1996), p. 123-155<http://www.numdam.org/item?id=M2AN_1996__30_2_123_0>

© AFCET, 1996, tous droits réservés.

L’accès aux archives de la revue « Modélisation mathématique et analysenumérique » implique l’accord avec les conditions générales d’utilisation(http://www.numdam.org/legal.php). Toute utilisation commerciale ou impres-sion systématique est constitutive d’une infraction pénale. Toute copie ou im-pression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiques

http://www.numdam.org/

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i ü J MATHEMATICAL MODEWNG AND NUMERICAL ANALYSISB B f l MOOÉUSATWN MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

(Vol. 30, n° 2, 1996, p. 123 à 155)

ON THE CONVERGENCE RATE OF SPECTRAL APPROXIMATION FORTHE EQUATIONS FOR NONHOMOGENEOUS ASYMMETRIC FLUIDS (*)

by José Luiz BOLDRINI C1) and Marko ROJAS-MEDAR C1)

Abstract. — We study the convergence rate of solutions of spectral semi-Galerkin approxima-tions for the équations for the motion of a nonhomogeneous incompressible asymmetrie fluid ina bounded domain. We find error estimâtes that are optimal in the H -norm as well as improved

Résumé. — On étudie le taux de convergence d'une approximation de type semi-Galerkinspectrale vers la solution des équations du mouvement d'un fluide assymétrique incompressiblenon-homogène dans un domaine borné. On trouve des estimations d'erreur qui sont optimalesdans la norme H ainsi que des estimations améliorées dans la norme L .

1. INTRODUCTION

In this paper we will study the convergence rate of solutions of spectralsemi-Galerkin approximations for the équations for the motion of a nonho-mogeneous viscous incompressible asymmetrie fluid. These équations areconsidérée! in a bounded domain Q <= RM, n = 2 or 3, with boundary F, in atime interval [0,7] , To describe them let u(x, t) e R", w(x, t) G R",p(x,t)e R and p(x, t) G R dénote, respectively, the unknown velocity,angular velocity of rotation of the fluid particles, the density and the pressureat a point x G Q, at a time t G [0, T1]. Then, the governing équations are

p + p( u . V ) M - (/i + /jr) Au + gradp = 2 }ir rot w + pf,

div u = 0 ,

p ^ + (M .V)W-(Ca+C r f)AW-(C0+C</-Ca)Vdivw (1.1)

+ 4 fir w = 2 pr rot u + pQ ,

Manuscript received February 11, 1993 ; revised March7, 1994.C1) UNICAMP-IMECC ; Caixa Postal 6065 ; 13081-970, Campinas, SP, Brazil.

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124 José Luiz BOLDRINI, Marko ROJAS-MEDAR

together with the following boundary and initial conditions

u = 0 on fx(OJ),

u(x,O) = uo(x) in Q,

w = 0 on r x ( O , r ) , (1.2)

^p(x,O°)=po(x) in D.

where, for simplicity of exposition we have taken homogeneous boundaryconditions.

Hère f(x, t) and g(x, t ) are respectively known external sources of linearand angular momentum of partiales. The positive constants ju, jjr, Co, Cc,Cd characterize isotropic properties of the fluid ; /u is the usual Newtonianviscocity ; jur, Co, Ca, Cd are new viscosities related to the asymmetry of thestress tensor, and in conséquence related to the appearance of the field ofinternai rotation w ; these constants satisfy Co+ Cd> Ca. The expressionsgrad, A, div and rot dénote the gradient, Laplacian, divergence and rotational

operators, respectively (we also dénote the gradient by V and — by ut) ; thef-th component of ( u . V ) u and ( u . V ) w in cartesian coordinates are givenby

du n dw

«;^i and [(«.V)H=2«;âT

n 3

respectively ; also ( u . V ) p = 2 w, " .

For the dérivation and physical discussion of équations (1.1) see Petrosyan[9] and Condiff, Dahler [2]. We observe that this model of fluid includes asa particular case the classical Navier-Stokes, which has been much studied(see, for instance, the classical books by Ladyzhenskaya [4] and Temam [15]and the références there in). ït also includes the reduced model of thenonhomogeneous Navier-Stokes équations, which has been less studied thanthe previous case (see for instance Simon [14], Kim [3], Ladyzhenskaya andSolonnikov [5] and Salvi [13]).

Concerning the generalized model of fluids considered in this paper,Lukaszewicz [8] stablished the local existence of weak solutions for (1.1),(1.2) under certain assumptions by using linearization and an almost fixed

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUTDS 125

point theorem. In that same paper Lukaszewicz remarked about the possibilityof proving the existence of strong solutions (under stronger hypothesis) byusing the techniques of [6] and [7] (linearization and fixed point theorems ; [6]and [7] assume constant density).

More interested in techniques directly related with numerical applications,Boldrini and Rojas-Medar [1] established the local (and also global) existenceof strong solutions of (1.1), (1.2) by using the spectral semi-Galerkin method(see Boldrini and Rojas-Medar [1] and also the next section for the précisestatements of the results). Hère, the word spectral is used in the sensé that theeigenfunctions of the associated Stokes and Laplacian operators are used asthe approximation basis.

Since Galerkin methods are much used in numerical simulations, it isimportant to dérive error estimâtes for them, even in the case of spectralGalerkin method, as a préparation and guide for the more practical finiteelement Galerkin method.

In this paper we are interested in establishing such error estimâtes and theconvergence rates of these spectral approximations in several norms. But,before we describe our results, let us briefly comment related results.

Rautmann in [10] gave a systematic development of error estimâtes for thespectral Galerkin approximations for the solutions of the classical Navier-Stokes équations. Salvi in [12] gave analogous error estimâtes for the reducedmodel of nonhomogeneous viscous incompressible fluids. However, althoughthe statement of Theorem 3, p. 203, in [12] furnishes an optimal rate(Kilv w n e r e An+1 is the (n + 1 )-th eigenvalue of the Stokes operator), thisis not correct as it can be seen by the last inequality in the proof (p. 204 in[12]). The rate actually obtained was 1~+

1/2.

In this paper we consider the convergence rate of the spectral semi-Galerkinapproximations for the solutions of the more gênerai fluid model (1.1). Weshow that there is optimal rate of convergence in the H1-norm (see Theo-rem 3.3), improving in the particular case of the reduced model the resuit inSalvi [12]. Differently as in the case of the classical Navier-Stokes équations,for which optimal L2-error estimâtes can be obtained (see Rojas-Medar andBoldrini [11]), in this case we are only able to obtain an improved L2-errorestimâtes as compared to the trivial one that dérives directly from theH'-estimate (see Theorem 4.2). Also, LT and heigher order error estimâtes areproved (see Theorems 3.4 and 3.5).

Finally, we would like to mention that optimal H1 and L2-error estimâtes forspectral Galerkin approximations for the Boussinesq and magnetohydrody-namic type équations can be obtained. These results will appear elsewhere.

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126 José Luiz BOLDRINI, Marko ROJAS-MEDAR

2. PRELIMINARIES

Let Q a M", n = 2 or 3, be a bounded domain with smooth boundary

(class C 3 is enough).

We will consider the usual Sobolev spaces

m = 0,1,2,..., l ^ ig f^+oo , D = & or 12 x (0,7) ,0 < T < + ©os with the usual norm. When q — 2, we dénote byHm(D) = W2'*(Z>) and ffJ(D) = closure of C~(D) in Hm(D). If B is aBanach space, we dénote by Lq( [0, r ] ) ; B) the Banach space of the Z?-valuedfunctions defined in the interval [0, T] that are L9-integrable in the sensé ofBochner. We shall consider the following spaces of divergence free functions

Cla(Q) = {ve(CZ(Q))n;divv=Q in Q],

H= closure o f C £ f f ( Ö ) in ( 2 n

V = closure ofCj f f(fl) in (Hl(Q))n .

Throughout the paper, P dénotes the orthogonal projection from(L2(Q))n onto H and A = - PA is the Stokes operator. We will ^enoterespectively by <p and kk the eigenfunctions and eigenvalues of the Stokesoperator defined in V n (H2(Q))n, It is well known that {<pk(x)}^=l form aorthogonal complete system in the spaces H, V and V n (H2(Q))n with theusual inner product, ( M, t? ), ( Vw, Vü ) and ( P Au, P Av ), respectively. Hère(.,. ) dénotes the inner product in L2(Q) ; also in this paper we will dénotethe L2-norm by || ||.

For each k G N, we dénote Pk the orthogonal projection from (L2(Q))n

onto Vk = span [<pv ..., <pk].It is easy to see that P and Pk, Pm, k, m G M satisfy for ail

ƒ, 0 e ( 2 n

(ƒ>ƒ,</) = (ƒ,/></)

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 127

The folio wing results can be found in Rautmann's paper [10]. If v e V, thenthere holds

(2.1)

Also, if v e Vn (H2(Q))n, we have

\v-Pkv\\2*-±-\\PAv\\2 (2.2)

\\Vv -VPkv\\2 t* j l - \\P Av\\2 . (2.3)

Now we observe that if ƒ e ( t f 1 ^ ) ) " , from (2.2) we have

|| U-Pk) Pf\\2 j 1 - II VPff . (2.4)Ak+l

Also, since P : (Hx(Q))n —» (Hl(Q))n is a continuous operator (see[16]), we have

| |VP/ | | 2 ^C| | / | | ^ . (2.5)

Thus, for ail ƒ e (Hl(Q))\ we have

II ( / - p , ) / y II < T^- i l ƒ ii^ , (2.6)A J k + l

or equivalently, since PPk = PkP = PkJ we obtain

WPf-Pjf^j^- \\f\\2Hl . (2.7)

We observe that (2.4)-(2.7) also holds with any Pm, m> k, in place of PAnalogously, we have for any ƒ e (£f2(Q))n

\\(I- Pk) Pf\\2 -f- | | / | | ^ . (2.8)

An easy conséquence of the L2-orthogonality of the {<pk}^=l is the follow-ing : let m > Jfc, m, A: G M, ƒ G (L 2 ( i2) )" and i?m G Vm, ü t e Vfc ; then

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128 José Luiz BOLDRINI, Marko ROJAS-MEDAR

Now, let us dénote B = - A : D(B) c (L2(Û))n -» (L2(Q))n whereD(B) dénotes the domain of - A with the Dirichlet boundary conditions and<f> ( x ) , <xk be the eigen-functions and eigen values of Z?, respectively. As it is> (well known, all the above properties have a corresponding one for £.

We will dénote Rk, k e N, the orthogonal projection of (L2(Q))n ontospan [<pv ..., 4>k].

It will be also necessary the following variant of the Gronwall's inequality(see Rautmann [10]).

LEMMA 2.1 : Let a function a(t) ^ 0 be absolutely continuons witha\t) 2* 0 and b{t) ^ 0 summable in [0, T]. Assume the intégral inequality

Jo

holds for the positive continuons functions £ and f on [O, T] with a constant

Then we have

with

Concerning the existence of solutions for équations (1.1), (1.2), they can beobtained by using a semi-Galerkin approximation, That is, we consider aGalerkin approximations

= SM2 AN Modélisation mathématique et Analyse numérique

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 129

for the velocity and rotation of particles respectively and an infinité dimen-siona]tionssional approximation pk(x, t) for the density satisfying the following equa-

fPk(pk ut + pk u Vu - pkf~ 2 fur rot wk) + (p + nr) Au = 0 ,

Rk( ph wk

t + pk u Vwk - pk g - 2 nr rot u - ( Co + Cd - Ca ) V div wk

u\0)=Pkuo,

w\O)=Rkwo,

[pk(0)=po.(2.10)

As before Pk and Rk are the orthogonal projections onto the spaces spannedby [<pv ..., cpk) and {01,..., <f>k}, respectively.

It can be proved that (uk,wk,pk) converges in appropriate sensé to asolution (M, w,/?) of (1.1), (1.2). As we said in the Introduction, in this paperwe are interested in deriving error bounds, that is, estimâtes forII « - uk ||, || w - wk ||, || p - pk || in suitable norms in terms of powers ofl/ ; and 1/ .

These error estimâtes will be derived in the following sections and will bebased on the next resuit. To easy the notation, in the rest of this paper thefunctions which are IR or Rn valued will not be notationally distinguished ; thedistinction will be clear from the context.

THEOREM 2.2 : {Boldrini and Rojas-Medar[l]). Let the initial values satisfy«oe V n (H2(Q))n, w0 e H\{Q) r^H2{Q), poe Wh~(Q) and the exter-nalfieldsfige L2(0, T; H\Q)) withft, gte L2(0, 7 ; L2(Q)). Then, on a(possibly small) time interval [0, T] the problem (LI) and (1.2) has a uniquestrong solution (u,w,p). That is, there are functions u, w, p such that

P(put + puVu - 2fiirrotw -pf- (JLI + jur) Au) =0

holds a.e. in Q x [0, T] ;

pwt + pu Vw ~-2firrotu-pg-(Ca + Cd) Aw -

-(C0+Crf-Cfl)Vdivw

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130 José Luiz BOLDRINI, Marko ROJAS-MEDAR

holds a.e. in Q x [0, 7] ;

holds in the L2(Q x [0 ,7] ) sensé. Moreover,

p<= Wh°°(Qx [ 0 , 7 ] ) ,

M G C([0, 7] ;//2(i3)nV)nL2(0J;//3(Û))nL2([0,r] ;L°°(i3))

u,e C([0,7] ;L2(f2)n V)nL2(0, 7; H2'E(Q))nLp( [0,7] ;Hl~\Q))

n L'(0, 7; H^~\Q)) n ^o c((0, 7]

wte C([0,7] ; 2 ^ 2

^ „ ^ L2OC(0,7;L2(£>))

for all e > 0 and 1 < p < + «>.

Remark : Actually it is possibly to prove that the strong solution ofTheorem 2.2 is global either if n = 2 or if we take small enough initial datawhen n = 3 (Boldrini and Rojas-Medar [1]).

The above resuit dépends on the certain estimâtes for the approximations( u\ w*, pk\ and since these estimâtes will be also necessary in this paper, wedescribe them in the following.

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 131

LEMMA 2.3 : {Boldrini and Rojas-Medar [1]). Let ( w\ wk, ph) be the solu-tion of (2,8), Then, they satisfies

\'{\\Aw\s)\\2+\\PAuk(s)\\2}ds^F2(t),Jo

\'{\\wk(s)\\2 + \\uk(s)\\2} ds F3(t) ,Jo

k(t)\\2+ ||«*(r)||2+ f '{ | |V^( 5 ) | | 2+ \\Vuk(s)\\2\\wk(t)\\2+ ||«*(r)||2+ f { | | V ^ ( 5 ) | | 2 + \\Vuk(s)\Jo

+ \\div

\\PAuk(t)\\2+\\Aw\t)\\2^F5(t),-

Ï{\\uk(s)\\2H, + \\W\s)\\2

Hi}ds*ZF6(t),Jo

\'{\\Vuk(S)\\2

L- + \\Vw\s)\\l~} ds F7(t) ,Jo

<y(t) {IIVK*(01|l- + | |Vv / (0 I I l - } < F1 0( t) ,

l'c7(s){\\PAuk(s)\\2+\\Awkt(s)\\

2}ds^Fn{t)>Jo

a =S pk ^ P, (0 < a = ess inf p0, fi = ess sup p0)

IIVAOII2- «F 1 2 (O,

11/ (0 II L- 'SfisCO,

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132 José Luiz BOLDRINI, Marko ROJAS-MEDAR

Hère, a(t) = min{l, t}.The same estimâtes hold for (w, W,p).In the following we assume the bounds on the right-hand sides of the above

estimâtes are chosen in such a way that they are monotonously increasing intime.

Remark : The above estimâtes implies that the approximations(w\ w\p k) converges to the solution ( M, w\ /? ) in the sensés indicated below

(i) w*->w, w*->w strongly in LP(O,T; H3~£(Q)) and weakly-* in3

oc

(ii) uk -> uv wk-^wt weakly in L£,c(0, T; H\Q) ) and weakly inL2(0,T;H2-£(Q)) inLp{O9T\H1-\Q)) and in L^c( 0, T ; H2( Q ) )

(iii) M* -> utt, wktt -> wtt weakly in L*oc( 0, 7 ; H2{ Q ) )

(iv) pk->p strongly in Lp(0, T; COy(Q)) 0 ^ y < 1(v) V / -^ Vp weakly-* in L°°(Q x [0, 7] )(vi) />?-»/>, weakly-* in LT(Q x [0, T] ).Hère, as before, the above is true for all e > 0 and 1 <p < + oo.Finally, we would like to say that as is usual we will dénote by C a generic

constant depending at most on Q and the fixed parameters in the problem(ju, jjLr, Ca, Cd, Co and the initial conditions, and also ƒ, g and T). This willappear in most of the estimâtes to the be obtained. When for any reason wewant to emphasize the dependence of a certain constant on a given parameterwe will dénote this constant with a subscript.

3. ERROR BOUNDS FOR THE APPROXIMATIONS

Let [0,7] be a time interval as in Theorem 2.2; u, w*, pk the &-thapproximations of w, w, p respectively. We begin by considering the following.

THEOREM 3.1 : Suppose the assumptions of Theorem 2,2 hold. Then, theapproximations u , w , p satisfy

Ak+\(3.1)

f ' | |V i« ( j ) -iw(f)-v/(OU2+ f'||Vi«(j)-Vu*(s)||2<to

+ f \\Vw(s)-Vw\s)\\2ds+ f ||div w(s) - div wk(s)\\2 dsJo Jo

r 7 ) (3-2)

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 133

for any te [0, 7 ] . The continuons fonctions Go( t),Gx(t) depend on t and onthe functions Ft(t) in Lemma 2.3. (3.1) and (3.2) hold also with any wm,wm, pm instead of u, w, p for m > k.

Proof: First we suppose (3.2) true. The différence pm — ph with m>ksatisfies

and

Po-Po = Q-

Let zm(t,s,x) be the solution of the Cauchy problem

z? = um(zm,s)

zm = x for t = s .

Then, by using the characteristic method, we obtain

pmU t) - p\x, t)=- \(pm k(zmU ï, Je), s) ds

where

<Pm,ktzm, t) = (um(zm, t) - u\zm, t) )Vp\zm, t) .

Bearing in mind properties of z" (see [5, pp. 93-96]), we get

\\pm-pk\\ ^ f ||Mm-M*|| \\Vpk\\L-ds < C f ||Mm-«*|| dsJo Jo

thanks to the estimâtes in Lemma 2.3. Hence by taking the limit as m goes toinfinity (see the Remark after Lemma 2.3), we get

\\P-Pk\\ ^C\ \\u-uk\\ds.Jo

Consequently, by using (3.2), we get (3.1).

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134 José Luiz BOLDR1NI, Marko ROJAS-MEDAR

Now, we prove (3.2). We consider the following équations (m>k)

- (CQ + Cd-Ca)RmV div wm = 0 (3.3)

Rk{pk wk + pk uk Vw* - pk g + 4 /Jr wk - 2 nr rot uk) + ( Ca + Cd) Bwk

-(Co+Cd-Ca)RkVdivwk = O. (3.4)

Subtracting (3.3) from (3.4), the différences

£ = wm-w* and ri = um-uk

satisfy

RmPm W7 ~ R

kPk w\ + R

mPm "m VH/" k * *

- 4 /JrRkwk- 2 nrRmvot um + 2 vrRkrotuk - (Co + Cd~Ca)RmV div wm

We take the inner product in L2( Q ) of (3.5) with £, after some computation,we obtain

(Ca+Cd)

V div v/) , O + (Rm(pm -pk)(g + wk + um VvO, {) + \ (p™ t\

r (3.6)

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 135

By using the Young's inequality, we get

+ C-eU\\2 . (3.7)

Also, we observe that

\(Rm(pm -Pk) (g + w] + um

(3.8)

Consequently, by virtue the estimâtes of Theorem 2.2, and (2.7), we get

ci' ||»7l|2<fa{CB||0||^ +CJVwf||2+Ce||PA«m||2||Awm||2}

. (3.9)

Moreover, bearing in mind the property (2.9) we have

• (C 0 +C r f -C f l ) Vdivw*,O|

l/Vl|w*|| + 2yur CHVM^H + 0||wf|| +fiU\\ +P\\ukVwk\\

+ (c +c - c ^ i i v d - *m M w > "

^ o </ ^11 l v w H) Πt + i

^ - (3.10)by virtue the estimâtes in Lemma 2.2.

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136 José Luiz BOLDRINI, Marko ROJAS-MEDAR

Now, we have

(3.11)

again thanks to the estimâtes in Lemma 2.3.From the estimâtes (3.7)-(3.11) we get the differential inequality :

*

By integrating this last inequality, we get for any ë > 0

(Pm)mtII2 +(Ca + Cd) f' | | V f | | 2 ^ + 4 iu f' l l ^ l l 2 ^Jo Jo

+ (C0+Cd-Ca) f Hdiv^ll2^Jo

|2dJa+ |"||I Jo

ak+i Jo

\\\\tt\\2+ U\\2}ds + -fï-+ ||^/2^0||2 + £ f

Jo Œk+l Jo(3.12)

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 137

in virtue of the estimâtes in Lemma 2.3.

Similarly, for r\ = um - uk, we have for any ö > 0

^k + 1

\\V^\\2ds. (3.13)

Adding inequalities (3.12) and (3.13), and taking ë > 0 and ö > 0 in suchway that (C a + Cd) -ô> 0 and / + ^ / r - ë > 0, we obtain the intégralinequality

f ||diJo

Ak+l ak+l

We observe that

xjfc+l

and

Ak+l

Using (3.15) and (3.16) in (3.14), we get :

iiv/7ii2}^+c2 rJo

C3Jo l^Jt+i "-t+i

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(3.14)

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138 José Luiz BOLDRINI, Marko ROJAS-MEDAR

Now applying Gronwall's inequality (Lemma 2.1) we obtain

r{iiv^n2+iivi7ii2}&+c2 r

C exp( C3

Now, by taking the limit on m goes to infinity (using the Remark afterLemma 2.3) on the left side we obtain (3.2). Thus Theorem3.1 proved. •

LEMMA 3.2 : Under the hypotheses of Theorem 2.2, the approximation pk

satisfy :

I I />(O-AOIIL ' ^ G2(0\^—^Y-\ (3.17)

with 2 ^ r ^ 6 for any t e [0, T]. The continuous function G2(t) dependon t and on the functions Ft(t) in Lemma 2.3. Also, (3.17) holds with anypm instead of p for m 5= k.

Proof: First, we observe that since Q is a bounded domain,UX{Q) czLr2(Q) with continuous inclusions if rx ^ r2 ^ 1. Therefore, itis enough to take r G R such that 3 ^ r ^ 6. Let m > k, m, k e N ; then

p™ + umVpm = 0 (3.18)

/>; + u * V = 0 (3.19)

Subtracting (3.19) from (3.18), the différence n = pm-pk satisfies

nt = - n V//" - / VTT

7l(0)=0

where tj = um — u . Now, multiplying by | n \r ~ and integrating over Q, weget

jj\\n\\r = jjïVpm\n\r-1 dx

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 139

thus,

since 3 ^ r ^ 6 the estimâtes after formula (3.16) furnishes

II « II i- ^c\\\Vr,\\2ds^CGl{t)\-±-+1±-\.

Thus, we have

\\pm(t)-pk(t)\\2Lr

k+i

Finally by taking the limit as m goes to infinity (see the Remark afterLemma 2.3) on the left side, we obtain (3.17). Thus, the Lemma is proved. •

Now, we have

THEOREM 3.3 : Under the hypotheses of Theorem 1, we have

+ f \\™t-™t\\2ds ^ CG 3 (O(- J -+T i -1 (3.20)

for any t G [0, T]. The continuons fonctions G3(t) dépends on t and on thefonctions F((t) of Lemma 2.3. Also, (3.20) holds with any um, wm instead ofu, w for m > k.

Proof: By taking the inner product in L2(Q) of (3.5) with £,, after somecomputations, we obtain :

- * * ) ( - / wk, + 2nrrot uk + pkg- 4 nrwk

(C0+Cd-Ca)Vdivwk

+ (Rm(- 2/irrot tj - 4tirt + pk{Vwm+pkukVi, £,)) . (3.21)

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140 José Luiz BOLDRINI, Marko ROJAS-MEDAR

Now, we observe that

+ (C0+Cd-Ca)V div wk-pkukVwk\ (t)\

CI I ^ + 4 Ai? Il rot u* | |^ + \\pkg\\2

H*

(3.22)

thanks to the estimâtes of Lemma 2.3 and 2.7. Also,

\(Rm(pm - p") (-wht + g-um Vwm), O\

J - + T J " } { C + IIVwf||2} + eKJ|2 (3.23)

in virtue of Lemma 3.2 and again the estimâtes of Lemma 2.3. Similarly weget

+ e||É,|r. (3.24)

Now, we observe that

and by taking e = a/6, we get from (3.21)

j fc+1

k+x Ak

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 141

This differential inequality yields the intégral inequality

f ut\\2ds+\\vç\\2

Jo

t+ C2 \\\Mfds + C3 f Hw*|| 3 dso

f Hw*||^3Jo

T J - + T M \ f \\Vwkt\\ak+i Ajt+ij Uods

which, together with

2

the resuit of Theorem 3.1 and the estimâtes in Lemma 2.3 yields the statedresuit for vu The resuit for u can be obtained analogously. •

THEOREM 3.4 : Under the hypothesis of Theorem 2.2, we have

(3.25)f'Jo

f'Jo

(3.26)

\\pU)-At)\\2L. < L3('){^7+iM (3 2 7>

for any ? G [ O, T~\. The fonctions continuous Lt( t ) depend on t and thefonctions F.(r) in Lemma 23. (3.25), (3.26) and (3.27) hold with any um,wm, pm instead of w, w, p for m> k.

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142 José Luiz BOLDRINI, Marko ROJAS-MEDAR

Proof : First we suppose (3.26) and we prove (3.27). We have

\\p-pk\\l~

Consequently,

\\p-pk\\l-

Csup | jV/ | | 2 - \\\PAu-PAuk\\2dsJo

bearing in mind (3.26).

To prove (3.26), we consider the following équations with m > k :

Pm(pm u™ + p m um Vum - pmf- 2 fxr rot wm - (ju + jjr) Aum ) = 0 (3.28)

Pk(pk ut + pk u Vu -phf-2 jur rot wk - (/i + nr ) ÀM* ) = 0 . (3.29)

Subtracting (3.29) from (3.28) and taking the inner product in L2( Q ) of theresuit with — P Arj, we get

^ (Pm(pmrit + pmumVij + pmr,Vuk-2Mrrott,

+ (P(pmP") (u + ukVukf)

-pkf,PAri)

\\pmumV«\\2+ \\pmrjVuk\\2

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 143

Thus, by taking e = (/i +/ir)/2 and estimating the tenus in the right-handside using the results of Lemma 3.2 we have

Ak+1

r4Ak+i

(we recall that C dépends on the functions F((t) of Lemma 3.2 and T),

By integrating the above inequality of 0 a t we get

Analogously we prove (3.25). This complete the proof of Theorem 3.4.

Finally, we have

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144 José Luiz BOLDRINI, Marko ROJAS-MEDAR

THEOREM 3.5 : JJnder the hypotheses of Theorem 2*2, the approximations

pkf u, wk satisfy :

<T(t)\\(pk)m(ut-ukt)f+ fais) ||Vu,-Vi£||2<fc^

Jo

vit) Il(/)1 /2(w f-^)| |2+ i'*(s) \\Vwt-Vwkt\\

2dsJö

a{t) \ ^[ak+\

a{t) ||Aw-Aw*||2^L1(r

ait) || M-M* || J- ^ CLit)

ait) ||W-w*||^ ^ CL.it) \-±

f a(s) | |VM-V/||^^Jo

Ak+l

k+i

} 1 {

j

where O a < 1/4 and o(t) = min{l, t}.

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 145

If the dimension is two, we have that for all e > 0, there is Ce > 0 :

fer<0 | | V w - V w * | | * - ^ CeAf2(

for every f e [0, T]. The continuom functions lî(f), *i(O»Af(r), M^t), N(î) and iV^t) depend on r and on the functions F.(r) inLemma 2.3. The above estimâtes holds also with any um, wm instead of w, w,m > k.

Proof : We only proof the first two estimâtes ; the others are provedsimilarly.

Differentiating (3.28) and (3.29) with respect to f, subtracting the results andtaking the inner product in L2( Q ) with r\t, we obtain

where

The above inequality implies

+ 11/ II v II ««II ll(/-^)<ll+2iur||rotwni Il(/-P,)«ril+ 2/ir||rot£f|

C(t) ||^||2 +ak+\

f ( « > 0 ) (3.30)

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146 José Luiz BOLDRINI, Marko ROJAS-MEDAR

where we use the young inequality (2.9), (2.1), Lemma 2.3 and Theorem 3.4.Analogously, for every 3 > 0 we have

(Ca + Cd) \\VÇt\\2+ (C0 + Cd-Ca) ||div£

, 1 d M / k\V2 * M 2

(3-31)"Jfc+1 " J t + 1

where

h(t)=Rkpkwk

t- Rmpm < + /? , ( / uk Vwk

Adding the inequalities (3.30) and (3.31) and taking e > 0 and S > 0 sothat (Ca+ Cd)-e>0 and (n + /ur) -S > 0, we get

xk+)

«^||Vdivw*|)2+

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 147

Multiplying the above inequality by a(t) = min {1, t] integrating of O at, using the Theorem 3.3 and Lemma 2.3 we obtain

O f W ) l|Vï/f||2ifa+ (Ca+Cd) fais) \\VÇt\\

2ds (3.32)Jo Jo

(C0+Cd-Ca) fais) ||divÉ,||2<fc + 4Air Î a(s) Ut\\2ds

Jo Jo

since <7(0)||(/>*)1'2>7,||2 = 0 and <T(0)

Now, we estimate the first intégral on the hand-right side of the aboveinequality ; the second intégral can be done similarly. We observe that

\ta(s)\\(Z(s))\\ \\r?t\\ dsJo

\2ds\ Ij'^s) \\rf\\2\\\(Z(s))\\2ds\ I j ^ s ) \\rft\\2ds

a(s)

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A yJ

1/2

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148 José Luiz BOLDRINI, Marko ROJAS-MEDAR

Hère we used (2.2) and the estimâtes of the Lemma 2.3. Now, we will provethat the intégral on the hand-right side of the above inequality is finite. Thisis proved of in a standard manner by using the estimâtes of the Lemma 2.3.Consequently in (3.32), we have

\\VÇt\\2 ds

\\div Zt\\2 ds + 4 nr \o(s) \\Zt\\

2ds/o

Ak+1

Now, by taking the limit as m goes to infinity (using the remark afterLemma 2.3), on the left side we obtain the first two estimâtes of the Theo-rem. •

4. EMPROVED L2 -ERROR BOUNDS

The L2-estimates obtained in Theorem 3.1 are not optimal ; in fact it is

expected to obtain a rate of convergence of order — 1—— instead of

K aonly Y1— + —*—.

Ak+\ ak+\

We were not able to do that, but in this Section we will improve theL2-estimates in Theorem 3.1 by using a bootstrap argument.

In order to do that, let u = 2 at(t) <p\x) and w = 2 bt(t) <p\x) be the

eigenfunction expansions of u and w. Let u = 2 a(t) (p\x) and

zk = 2 b(t) <pl{x) be the £-th partial sum of the series for u and w,

respectively, and let

e = M - i?*, e* = w - z \ y/k = uk - vk and ak = wk -zk ,

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 149

where uk and wk be the &-th Galerkin approximations of u and M>, respectively.Then, we have u - u = ek - y/k and w - wk = ek - ak.

With these notations, we state

LEMMA 4.1 :

f ( l lV | | 2 + ||V</||2 + ||divc^||2)ifa

7Ü2~+ 1/2 i fak+îAk+l ak+lAk+lJ

^ C(t) -1-2 +~2 +"-3^2~+^372~+ 7Ü2lak+l Ak+l ak+l Ak+l ak+îAk+

for any f e [0, 71]. 77ie continuous function C(t) depend on t and on thefunctions, Ft(t) in Lemma 23.

Proof: We observe that vk and z satisfy

Pk(put + puVu- pf- 2 fir rot w) + (fi + juf) Au* = 0 (4.1)

Rk{pwt + puVw- pg - 2/jrrotu)-h4firzk-

- (Co+ C r f- Ca)/?,Vdivw + (Cfl + Crf)&* = 0 (4.2)

for all, r e [0,7].Subtracting (4.1) of (2.8) (i) and (4.2) of (2.8) (ii) we obtain

(Ai + /ir) A¥k + Pk{pk uk -put) + Pk(p

k uk V / - pu Vu )

+ Pk(p - pk)f+2firPk(rot (w - wk)) = 0 , (4.3)

+ Rk(pk wk - pwt) + R ^ / w* Vw - pw Vw)

+ Rk(p - / ) g + 2 Airi?ft(rot (K - i#*)) = 0 (4.4)

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150 José Luiz BOLDRINI, Marko ROJAS-MEDAR

By taking the inner product in L2 of (4.3) with y/k and of (4.4) with ak andproceeding as in the previous Section, we obtain

+ ( (p - pk ) «* Vu*. / ) - ( / > < / VM*, y/)

+ (pek Vuk,y/k) - (pu Vy/k,y/k) + (pu Ve*,

+ (rot CT\ / ) - (£*, rot / ) + ((p - pk)f, / ) (4.5)

- (pek, ak) + ( Co + Cd - Ca) || div ak\\

«* Vwk, ak) + (puk, VÉ*. e/)

r*) + 2iur(rot^, ok)-2nr(ek, totak)

+ (C0 + Cd-Ca)(Vàiv?,ok) + ((p-pkg,Gk) ) . (4.6)

Also, we have

p-pk = - \\u-u")Vpk- l' ekVpk+ | V V .Jo Jo Jo

From this, we observe that if / is any function in (H\£2))n, for anyS > 0, there holds

d f l IvYl lVl l 2 . - \\x\\l*+2ô\\Vy,k\\2.

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 151

By using this with x = ƒ> X = u . Vw* and / = v\ and the following facts

for which we have used (2.2), together the estimâtes of the lemma 2.3 ;estimating as usual, (4.5) furnishes, for any ô > 0, y > 0,

en/H

<||2+ ||V(«iV«i)||2+ Il/Il2,,}

y\\V<jk\\2+l3S\\Vy,k\\2

| | / | |2<Ml|Vt>?| |2+ \\V(ukVuk)\\2+ Il/Il2,!}. (4.7)o

Proceeding in the some way with (4.6), we obtain for any Ö > 0,y > 0

(4.8)

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152 José Luiz BOLDRINI, Marko ROJAS-MEDAR

By adding (4.7) and (4.8), we are left with

^ HVdivaY

1 d M / Jfc\l /2 * | i 2 . 1 d M / ks.ï/2 k,,2 , k. 1 d M / ks.ï/2 k,,2 , k *-. / k K

2di"(p^ W " ~ (p<' a ^ ~ (pet>V )

a * + l Àk+\

xfe+i

Integrating with respect to t, we have (with Cj > 0 )

•kt,a

k)ds

. (4.9)

By intégration by parts with respect to t and recalling that ak{ 0 ) = 0, wecan estimate as follows

(pte\ak)ds

IJo r-ll^ll (II «fi

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 153

0

+ •xk+i ah

1/2

3/2 ,1/2

Similarly, we estimate

Thus, from (4.9) and the above, we are left with

n/(on2+ f( iiv/u2+ iivo*

2 y* 13/2 3/2 nl/2 2 1/2 I ''k+ï Ak+\ Ak+\ ait+l ak+\ Ak+\ Ak+l ak+\/

From this and Gronwall's inequality, we conclude that

11/(0 II2 + II<AOH2+ f( HV/(j)||2+ \\Vak(s)\\2ds)Jo

C € l —z h —z + —TT

\ 2 i2 yL

\ A J t + l Ak+\ AJtH

1 13/2 ^ 3/2 ^ n 1/2'Jt+1 ak+l ak+ 1 Ak+\ •^d

and the Lemma 4.1 is proved. •Now we are ready to prove the following

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154 José Luiz BOLDRINI, Marko ROJAS-MEDAR

THEOREM 4.2 :

* 2 + \\w(t)-w\t)\\2

m

*/ * \ II 2ii) \\p(t)-p\t)\\

/ o r any, î e [O, 71].

Proof : From estimâtes in Lemma 2.3 and (2.8) we conclude that

Ak+\ ^-k +1

Analogously, we have

Thus, since || w - u \\ = || e* - / 1 | ^ || ek \\ + || / 1 | and|| w - w* H = || ek -ak\\ ^ || ek \\ + || a* ||, Lemma 4.1 and the aboveimply (i).

Now, proceeding as we done in the begining of Theorem 3.1

\\p(t)~pk(t)\\2^C(t) \t\\u(s)-uk(s)\\2dS.Jo

This and (i) imply (ii). •We observe that we cannot obtain the optimal rate due to the terms pk ek and

pk ek. Ho wever, they do not appear in the classical Navier-S tokes équations,and also in the Boussinesq and mathetohydrodynamic type équations, and,consequently, for these équations it is possible to obtain optimal L2-errorestimâtes.

Remark : If we consider the reduced model of nonhomogeneous fluids, likein Salvi [12], the above estimâtes are reduced to

Ak+l Ak+l

with again of «• with respect to the convergence rate obtained in [12].

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SPECTRAL APPROXIMATION FOR NONHOMOGENEOUS FLUIDS 155

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[2] D. W. CONDIFF, J. S. DAHLER, 1964, Fluid mechanics aspects of antisymmetric

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[3] J. U. KIM, 1987, Weak solutions of an initial boundary value problem for anincompressible viscous fluid, SIAM J. Math. Anal., 18, pp. 890-96.

[4] O. A. LADYZHENSKAYA, 1969, The Mathematical Theory of Viscous Incompres-sible Flow, Gordon and Breach, Second Revised Edition, New York.

[5] O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, 1978, Unique solvability of an initialand boundary value problem for viscous incompressible fluids, Zap. Naucn Sent.Leningrado Otdel Math. Inst. Steklov, 52, 1975, pp. 52-109 ; English Transi., / .Soviet Math., 9, pp. 697-749.

[6] G. LUKASZEWICZ, 1988, On nonstationary flows of asymmetrie fluids, RendicontiAccademia Nazionale delle Scienze detta dei XL, Memorie di Matematica 106°>XII, fasc. 3, pp. 35-44.

[7] G. LUKASZEWICZ, 1989, On the existence, uniqueness and asymptotic propertiesof solutions of flows of asymmetrie fluids, Rendiconti Accademia Nazionale déliaScienze detta dei XL, Memorie di Matematica 107e\ XIII, fasc. 6, pp. 105-120.

[8] G. LUKASZEWICZ, 1990, On nonstationary flows of asymmetrie fluids, Math.

Methods Appl. Sri., 19, no. 3, pp. 219-232.

[9] L. G. PETROSYAN, Some Problems of Mechanics of Fluids with AntisymmetricStress Tensor, Erevan, 1984 (in Russian).

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Lecture Notes in Math., 771, Springer-Verlag.

[11] M. ROJAS-MEDAR, J. L. BOLDRINI, 1993, Spectral Galerkin approximations forthe Navier-Stokes Equations : uniform in time error estimâtes, Rev. Mat. Api., 14,pp. 1-12.

[12] R. SALVI, 1989, Error estimâtes for the spectral Galerkin approximations of the

solutions of Navier-Stokes type équation, Glasgow Math. J., 31, pp. 199-211.

[13] R. SALVI, 1991, The équations of viscous incompressible nonhomogeneous fluid :on the existence and regularity, 7. Australian Math. Soc, Series B - AppliedMathematics, 33, Part 1, pp. 94-110.

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